Question

Find the exact value of the given expression.

 sin(2cos^-1 24/25)

_______ 

Find the exact value of the given expression. 

sec(2 sin^-1 1/8)

_______ 

Answer 1

Answer 2

ANSWER :

9.

sin(2 cos^(-1) 24 / 25) 

Let  cos^(-1) 24 / 25 = x

=> cos(x) = 24/25  .

So, 

sin(2 cos^(-1) 24 / 25) 

= sin(2x) 

= 2 sin(x) cos(x) 

= 2 sqrt(1 - coś^2 x) coś(x)

= 2 sqrt( 1 - (24/25)^2) * 24 / 25

= 2 * sqrt((625-576) / 625) * 24 / 25

= 2 * sqrt(49 / 625) * 24 / 25

= 2* 7 / 25 * 24 / 25

= 336 / 625 (ANSWER).

10.

sec(2 sin^(-1) 1 / 8 )

Let sin((-1) 1 / 8 = x

=> 1/8 = sin(x)

=> cos(x) = sqrt( 1 - (1/8)^2) = sqrt(63 / 64)

=> cos^2 (x) = 63/64 

Now,

sec(2 sin^(-1) 1 / 8) 

= sec (2x) 

= 1/cos(2x) 

= 1 / (2cos^2 (x) - 1)

= 1 / (2*63/64 - 1)

= 1 / (62 / 64)

= 64/62

= 32/31 (ANSWER).